# Properties

 Label 9075.2.a.db Level $9075$ Weight $2$ Character orbit 9075.a Self dual yes Analytic conductor $72.464$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{3} ) q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( 2 + \beta_{3} ) q^{4} + \beta_{1} q^{6} + ( \beta_{1} - \beta_{2} ) q^{7} + ( \beta_{1} + 2 \beta_{2} ) q^{8} + q^{9} + ( 2 + \beta_{3} ) q^{12} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{13} + 2 q^{14} + ( 4 + \beta_{3} ) q^{16} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( \beta_{1} - 4 \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{21} -8 q^{23} + ( \beta_{1} + 2 \beta_{2} ) q^{24} + ( 2 - \beta_{3} ) q^{26} + q^{27} + 2 \beta_{2} q^{28} + 4 \beta_{1} q^{29} + \beta_{3} q^{31} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( 12 + 2 \beta_{3} ) q^{34} + ( 2 + \beta_{3} ) q^{36} -5 q^{37} + ( -4 - 3 \beta_{3} ) q^{38} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{39} + 4 \beta_{2} q^{41} + 2 q^{42} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{43} -8 \beta_{1} q^{46} + ( 4 + 2 \beta_{3} ) q^{47} + ( 4 + \beta_{3} ) q^{48} + ( -4 - \beta_{3} ) q^{49} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{51} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{52} + ( 6 - 2 \beta_{3} ) q^{53} + \beta_{1} q^{54} + 2 \beta_{3} q^{56} + ( \beta_{1} - 4 \beta_{2} ) q^{57} + ( 16 + 4 \beta_{3} ) q^{58} + 4 q^{59} + ( \beta_{1} + 2 \beta_{2} ) q^{61} + ( \beta_{1} + 2 \beta_{2} ) q^{62} + ( \beta_{1} - \beta_{2} ) q^{63} -\beta_{3} q^{64} + ( 4 + 3 \beta_{3} ) q^{67} + ( 6 \beta_{1} + 8 \beta_{2} ) q^{68} -8 q^{69} + ( 6 + 2 \beta_{3} ) q^{71} + ( \beta_{1} + 2 \beta_{2} ) q^{72} -3 \beta_{2} q^{73} -5 \beta_{1} q^{74} + ( -9 \beta_{1} + 2 \beta_{2} ) q^{76} + ( 2 - \beta_{3} ) q^{78} + ( -2 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 8 + 4 \beta_{3} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{83} + 2 \beta_{2} q^{84} + ( 18 + 3 \beta_{3} ) q^{86} + 4 \beta_{1} q^{87} + ( -4 + 4 \beta_{3} ) q^{89} + ( 7 - 3 \beta_{3} ) q^{91} + ( -16 - 8 \beta_{3} ) q^{92} + \beta_{3} q^{93} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{94} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{96} + ( 7 + \beta_{3} ) q^{97} + ( -5 \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} + 6 q^{4} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 14 q^{16} - 32 q^{23} + 10 q^{26} + 4 q^{27} - 2 q^{31} + 44 q^{34} + 6 q^{36} - 20 q^{37} - 10 q^{38} + 8 q^{42} + 12 q^{47} + 14 q^{48} - 14 q^{49} + 28 q^{53} - 4 q^{56} + 56 q^{58} + 16 q^{59} + 2 q^{64} + 10 q^{67} - 32 q^{69} + 20 q^{71} + 10 q^{78} + 4 q^{81} + 24 q^{82} + 66 q^{86} - 24 q^{89} + 34 q^{91} - 48 q^{92} - 2 q^{93} + 26 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 5 \beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.52434 −0.792287 0.792287 2.52434
−2.52434 1.00000 4.37228 0 −2.52434 −0.792287 −5.98844 1.00000 0
1.2 −0.792287 1.00000 −1.37228 0 −0.792287 −2.52434 2.67181 1.00000 0
1.3 0.792287 1.00000 −1.37228 0 0.792287 2.52434 −2.67181 1.00000 0
1.4 2.52434 1.00000 4.37228 0 2.52434 0.792287 5.98844 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9075.2.a.db yes 4
5.b even 2 1 9075.2.a.cw 4
11.b odd 2 1 inner 9075.2.a.db yes 4
55.d odd 2 1 9075.2.a.cw 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9075.2.a.cw 4 5.b even 2 1
9075.2.a.cw 4 55.d odd 2 1
9075.2.a.db yes 4 1.a even 1 1 trivial
9075.2.a.db yes 4 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9075))$$:

 $$T_{2}^{4} - 7 T_{2}^{2} + 4$$ $$T_{7}^{4} - 7 T_{7}^{2} + 4$$ $$T_{13}^{4} - 46 T_{13}^{2} + 1$$ $$T_{17}^{2} - 44$$ $$T_{19}^{4} - 79 T_{19}^{2} + 1156$$ $$T_{23} + 8$$ $$T_{37} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 7 T^{2} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$4 - 7 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$1 - 46 T^{2} + T^{4}$$
$17$ $$( -44 + T^{2} )^{2}$$
$19$ $$1156 - 79 T^{2} + T^{4}$$
$23$ $$( 8 + T )^{4}$$
$29$ $$1024 - 112 T^{2} + T^{4}$$
$31$ $$( -8 + T + T^{2} )^{2}$$
$37$ $$( 5 + T )^{4}$$
$41$ $$( -48 + T^{2} )^{2}$$
$43$ $$( -99 + T^{2} )^{2}$$
$47$ $$( -24 - 6 T + T^{2} )^{2}$$
$53$ $$( 16 - 14 T + T^{2} )^{2}$$
$59$ $$( -4 + T )^{4}$$
$61$ $$256 - 43 T^{2} + T^{4}$$
$67$ $$( -68 - 5 T + T^{2} )^{2}$$
$71$ $$( -8 - 10 T + T^{2} )^{2}$$
$73$ $$( -27 + T^{2} )^{2}$$
$79$ $$1 - 46 T^{2} + T^{4}$$
$83$ $$256 - 76 T^{2} + T^{4}$$
$89$ $$( -96 + 12 T + T^{2} )^{2}$$
$97$ $$( 34 - 13 T + T^{2} )^{2}$$